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10! and 10!+20

This topic has 2 expert replies and 1 member reply
j_shreyans Legendary Member Default Avatar
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10! and 10!+20

Post Fri Sep 05, 2014 10:39 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    How many integers are divisible by 3 between 10! and 10!+20 inclusive?

    A)6
    B)7
    C)8
    D)9
    E)10

    OAB

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    Post Fri Sep 05, 2014 11:37 pm
    Quote:
    How many integers between 10! and 10!+20, inclusive, are divisible by 3?

    6
    7
    8
    9
    10
    Two number property rules:
    (multiple of k) + (multiple of k) = (multiple of k).
    (multiple of k) + (non-multiple of k) = (non-multiple of k).

    Since 10! = 10*9*8*7*6*5*4*3*2, 10! is a multiple of 3.
    To yield subsequent multiples of 3, we must add to 10! consecutive multiples of 3.
    Options between 10! and 10! + 20, inclusive:
    10!
    10! + 3
    10! + 6
    10! + 9
    10! + 12
    10! + 15
    10! + 18.
    Total options = 7.

    The correct answer is B.

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    Post Sat Sep 06, 2014 6:03 am
    Quote:
    How many integers are divisible by 3 between 10! and 10!+20 inclusive?
    A) 6
    B) 7
    C) 8
    D) 9
    E) 10
    There's a nice rule that says: If M is divisible by k, and N is divisible by k, then (M + N) is divisible by k.
    Conversely, If M is divisible by k, and Q is NOT divisible by k, then (M + Q) is NOT divisible by k.

    First, since 10! = (10)(9)(8)..(3)(2)(1), we know that 10! is divisible by 3.

    So, by the above rule, we know that 10! + 3 is divisible by 3
    And 10! + 6 is divisible by 3
    10! + 9 is divisible by 3
    10! + 12 is divisible by 3
    10! + 15 is divisible by 3
    10! + 18 is divisible by 3

    So, there are 7 integers from 10! to 10! + 20 inclusive that are divisible by 3.

    Answer: B

    Cheers,
    Brent

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    Post Sat Sep 06, 2014 10:56 am
    Hi j_shreyans,

    The GMAT Quant section oftentimes tests you on math rules/concepts that you know, but in ways that you're not used to thinking about. This question is built on a division rule that you probably know: factoring.

    Here's a simple example of the concept:

    Is 10 + 8 divisible by 2?
    Yes; both 10 and 8 are divisible by 2, so adding them together gives us a bigger number that's divisible by 2.

    This can be proven by factoring:
    (10 + 8) = 2(5 + 4).

    With these ideas in mind, you can now tackle the prompt:

    10! = (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) so we know that this is divisible by lots of numbers. There are 21 integers between 10! and 10!+20 that we have to consider. Which of these integers is divisibly by 3?

    We already know that 10! is divisible by 3 (since there's a "3" that can be factored out). Now, what other numbers, when added to 10! are ALSO divisible by 3?

    10!
    10! + 3
    10! + 6
    10! + 9
    10! + 12
    10! + 15
    10! + 18

    Total integers = 7

    Final Answer: B

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